An investigation of the performance of parametric functional forms for the Lorenz curve

Given that an excellent performance of any parametric functional form for the Lorenz curve that is based on a single country case study and a limited range of distribution must be treated with great caution, this study investigates the performance of a single-parameter functional form proposed by Paul and Shankar (2020) who use income data of Australia to show that their functional form is superior to the other existing widely used functional forms considered in their study. By using both mathematical proof and empirical data of 40 countries around the world, this study demonstrates that Paul and Shankar (2020)’s functional form not only fails to fit the actual observations well but also is generally outperformed by the other popular functional forms considered in their study. Moreover, to overcome the limitation of the performance of a single-parameter functional form on the criterion of the estimated Gini index, this study employs a functional form that has more than one parameter in order to show that, by and large, it performs better than all popular single-parameter functional forms considered in Paul and Shankar (2020)’s study. Thus, before applying any functional form to estimate the Lorenz curve, policymakers should check if it could describe the shape of income distributions of different countries through the changes in parameter values and yield the values of the estimated Gini index that are close to their observed data. Using a functional form that does not fit the actual observations could adversely affect inequality measures and income distribution policies.

We sincerely thank Reviewer #1 for comment on this very important point. We would like to inform Reviewer #1 that we have made a substantial revision in Introduction by providing definition of the Lorenz curve in paragraph 1 and discussing how the Gini index can be calculated from the Lorenz curve in paragraph 2 in our revised manuscript. We also explain in Introduction, paragraph 3 in our revised manuscript how the Lorenz curve can be estimated as well as provide the list of up-to-date studies that propose different parametric functional forms for estimating the Lorenz curve.
-It is worth adding the relevance and purpose of the research.
When justifying its relevance, it is worth pointing out which economic and social processes are affected by income inequality, why is it important to study it?
For example: Impact of Income Distribution on Social and Economic Well-Being of the State https://doi.org/10.3390/su12010429 In response to Reviewer #1's suggestion to add the relevance and purpose of our study, we state the purpose of our study in Introduction, paragraph 4 in our revised manuscript which is to investigate an alternative single-parameter functional form for the Lorenz curve proposed by Paul and Shankar (2020) who show that their functional form outperforms the other 4 popular parametric functional forms for the Lorenz curve, namely, Kakwani and Podder (1973), Aggarwal (1984), Chotikapanich (1993, and a functional form implied by Pareto distribution. In addition, we state the relevance and contribution of our study in Introduction, paragraph 5 in our revised manuscript that it is relevant and worthwhile to conduct an investigation to find out if we use grouped income data of other countries, the performance of Paul and Shankar (2020)'s functional form is still superior to the other existing widely used functional forms considered in their study. The findings from this investigation should also contribute as a check-and-balance not only for economics but also for other scientific disciplines that use the Lorenz curve to analyze size distributions of non-negative quantities and inequalities.
Since the main focus of our study is to investigate the performance of parametric functional forms for estimating the Lorenz curve and to examine whether or not those parametric functional forms have an explicit mathematical solution for the Gini index, we provide an additional justification for our study as noted by Reviewer #1 and also by PLOS ONE Staff Editor Dr. Hanna Landenmark by noting in Introduction, paragraph 5 and in Conclusions, paragraph 3 in our revised manuscript that various studies on the relationship between inequality measures and financial and/or socioeconomic variables such as Kharlamova et al. (2018), Bilan et al. (2020), and Tung (2020) as suggested by Reviewer #1 and Pisular et al. (2013) as suggested by Reviewer #2, rely on the accuracy of inequality measures that could possibly be derived from a parametric functional form for the Lorenz curve. If the choice of parametric functional form is not a valid candidate for representing the income distribution, the estimates on the income shares and inequality measures might be severely affected by misspecification bias. This justification is reiterated in Conclusions, paragraph 3 in our revised manuscript. In response to the issue of investigating studies on the relationship between income inequality and other parameters as suggested by Reviewer #1, we went over those studies and find them very useful. However, given that the main focus of our study is to investigate the performance of parametric functional forms for estimating the Lorenz curve and to examine whether or not those parametric functional forms considered in Paul and Shankar (2020) study have a closed-form expression for the Gini index, we therefore include studies by Kharlamova et al. (2018), Bilan et al. (2020, Tung (2020) as suggested by Reviewer #1 and Pisular et al. (2013) as suggested by Reviewer #2 in Introduction, paragraph 5 in our revised manuscript in order to illustrate the importance of using the appropriate parametric functional form for estimating the Lorenz curve. We also mentioned the importance of this issue in Conclusions, paragraph 3 in our revised manuscript.

-Discussion should be separated into a separate section.
We follow Reviewer #1's suggestion by deleting "discussion" in our revised manuscript.

-It is worth noting other publications in which the Gini index is considered. For example: CAN PUBLIC DEBT HARM SOCIAL DEVELOPMENT? EVIDENCE FROM THE ASIAN-PACIFIC REGION doi:10.14254/2071-8330.2020/13-2/4
We follow Reviewer #1's suggestion by including study by Tung (2020) in Introduction, paragraph 5 and reiterating its relevance to our study in Conclusions, paragraph 3 in our revised manuscript.
-It is necessary to update according to the latest research, to issue according to the requirements.
We sincerely thank Reviewer #1 for pointing this out. We cite the up-to-date studies that propose different parametric functional forms for estimating the Lorenz curve in Introduction, paragraph 3 and include them in References in our revised manuscript.

Reviewer #2:
-The article is interesting in terms of topics, but not entirely correctly implemented in terms of structure. It is noticeable at the beginning that there is no review of world research in this area, i.e. the so-called Review of the literature. Throughout the article, the authors refer to only 15 items !!! This needs to be completely rewritten and corrected. Since you analyze, for example, the Gini index, it is worth referring to such items as: We sincerely thanks Reviewer #2 for comment and suggestion regarding the review of world research in the area of parametric functional forms for estimating the Lorenz curve. In our revised manuscript, we have made a major revision by citing the up-to-date studies in Introduction, paragraph 3 and including those studies in References. We also rewrite Introduction by providing the definition of the Lorenz curve and its applications in paragraph 1. In addition, we explain how the Gini index can be derived from the Lorenz curve in Introduction, paragraph 2. In Introduction, paragraph 5 and Conclusions, paragraph 3, studies by Pisula et al. (2013) as suggested by Reviewer #2 andKharlamova et al. (2018), Bilan et al. (2020), and Tung (2020) as suggested by Reviewer #1, all of which employ the Gini index in their analyses, are included as a part of our discussion on the importance of using an appropriate parametric functional form for estimating the Lorenz curve and calculating the Gini index.

Statistical methods of the bankruptcy prediction in the logistics sector in Poland and Slovakia, Transformations in Business and Economics
-Authors must also emphasize the so-called added value of research. What's new in these findings?
In response to Reviewer #2' s concern on the issue of the added value of our study, we state in Introduction, paragraph 5 in our revised manuscript that, given the superiority of Paul and Shankar (2020)'s functional form over the other existing widely used functional forms, it is therefore relevant and worthwhile to conduct an examination to find out if we use grouped income data of other countries, the performance of Paul and Shankar (2020)'s functional form is still superior to the other existing widely used functional forms considered in their study. The findings from our investigation should also contribute as a check-andbalance not only for economics but also for other scientific disciplines that use the Lorenz curve to analyze size distributions of non-negative quantities and inequalities. This is because using a functional form that does not fit the actual observations could adversely affect inequality measures and income distribution policies.
In addition to new findings reported in Results, we also summarize our main findings in Introduction, paragraphs 6 and 7 and in Conclusions, paragraphs 1 and 2 in our revised manuscript as suggested by Reviewer #2.